Discussion and Analysis of the Factors of an Electron Through
Measurements with Electron Spin Resonance
Kathryn Potter
Electron Spin Resonance (ESR) is a very important technique in
understanding physics principles at the atomic level. This technique allows
observations to be made for the electron’s spin and magnetic moment and
most importantly the electron g-factor. This factor connected the basic
principles of ESR to the complex principles behind relativistic quantum
mechanics. The connection to relativistic quantum mechanics allows an
observer to avoid all the underlying complications with making
measurements of an electron’s behavior. Therefore, ESR and the values it
observes and measured is crucial to understanding the physics concepts
occurring at the atomic level.
Understanding the laws of physics requires understanding objects and their
behaviors at an atomic level. One of these objects is the electron and its spin behavior and
magnetic moment. Measuring these factors is extremely rigorous though without
relativistic quantum mechanics. When relativistic quantum mechanics is introduced, a
factor of 2 is observed for the magnetic moment. The factor of 2 is called the electron gfactor and its value is calculated through the procedure in this paper. This calculation is
made by observing a magnetic moment in a static magnetic field precessing at a natural
frequency. Resonance can then occur when an oscillating external field is applied at that
frequency. It is from this resonance that the procedure makes certain observations and
measurements to calculate the g-factor. From this point forward, many advances were
made in regards to the electron and its spin component. One of these advances was made
when the affect of unpaired electrons in certain materials was measured. This new type of
measurement created the concept of electron spin resonance. The theory behind this
measurement method is described in the next section.
The theory behind the measurement method of the Electron Spin Resonance
(ESR) can only be described after first understanding the apparatus used.
Figure 1: ESR Diagram
The ESR apparatus includes a coil wrapped around a test sample, an RF Oscillator, and
an Ammeter. The test sample is placed in a uniform magnetic field and the RF oscillator
induces small magnetic fields in the coil at a right angle to the current uniform magnetic
field. These two magnetic fields create conditions for resonance, in which measurements
are made from. A single electron within the test sample has a spin, which gives it an
intrinsic magnetic dipole moment (µS). This relationship is seen in the equation
𝜇𝑆 = 𝑔𝑆 𝜇𝐵
𝑆
ℏ
(1)
In this equation 𝑔𝑆 is a constant characteristic of the electron (the g-factor); 𝜇𝐵 is the
Bohr magneton = 𝑒ℏ/2𝑚𝑒 = 5.788 x 10-9 eV/G (where e is the charge of an electron, me
is the mass of an electron, and ℏ is Planck’s constant = 6.582 x 10-16 eV·sec); and S is the
spin of the electron. The magnetic dipole moment of an electron is important because it is
this property of the electron that interacts with the uniform magnetic field.
A second property of the electron, which is briefly described above, is its spin of
± 1/2. Because of these two values for the spin, an electron can only have energies of
𝐸0 ± 𝑔𝑆 𝜇𝐵 /2, where 𝐸0 is the electron’s energy before it is placed in the uniform
magnetic field. The difference between these two possible energies gives a value of
𝑔𝑆 𝜇𝐵 B where B is the magnitude of the uniform magnetic field. It is these energy
differences, which creates the condition for resonance. Resonance occurs when the
frequency (𝜐) of the RF oscillator results in energy equal to 𝑔𝑆 𝜇𝐵 𝐵. The RF oscillator
irradiates photons at an energy of ℏ𝜐. At this energy electrons from a lower energy state
are able to jump to a higher energy state by absorbing a photon. When this occurs, the
test sample is affected, which in turn changes the coil inductance and the RF oscillator
oscillations. From these changes an observable change in current is observed and
measured. In summary, resonance occurs when the energy of the irradiated photons
equals the electron energy differences, which is seen in the equation
ℏ𝜐 = 𝑔𝑆 𝜇𝐵 𝐵
(2).
B in this equation is calculated from the following equation:
3
4 2
𝜇0 (5)
𝐵=
𝐼
𝑁𝑟
(3)
where 𝜇0 = 1.256 x 10-6 v·s/A·m; N is the number of turns in each coil (320); I is the
current through each coil; and r is the radius. Equation (2) can be solved for 𝑔𝑆 and
broken up more into its components in order to get the following equation:
𝜐
𝑔𝑆 = ( 𝐼 )
ℏ2𝑟
3
(4)
4
𝜇𝐵 𝜇0 ( )2 𝑁
5
However, actually achieving resonance is difficult and requires accurate measurement.
The measurement can only be taken when the RF frequency is set at exactly the
right value. This value is achieved by adjusting the value of the magnetic field about a
constant value. The PASCO ESR Apparatus shown in Figure 1 accomplishes this by
sending a small current to the coils around the test sample. This small current allows the
magnetic field to vary sinusoidally around a constant value. A dual trace oscilloscope
produces a sinusoidal wave demonstrating this value, as seen in Figure 2.
Figure 2: ESR on the Oscilloscope
The upper sine wave is a function of the current passing through the coils, which is
proportional to the magnetic field. The function below this sine wave demonstrates the
voltage passing through the RF oscillator. The dips in this function occur when the
magnetic field passes the resonance point. These two functions are the main components
in the measurement of the ESR.
The ESR measurements are still complicated due to various conditions within an
atom. These conditions result from complex electromagnetic factors and energy splittings
and shifts, which make the ESR measurements less accurate. However, with the PASCO
ESR Apparatus these conditions are simplified. The PASCO ESR Apparatus uses a
Diphenyl-Picryl-Hydrazyl (DPPH) test sample, which has a simple nature that makes
measuring the ESR easier and more accurate. The simple nature of DPPH is due to the
fact that is has an orbital angular momentum of zero and only one unpaired electron.
These two factors make measurement easier because now for any magnetic field there is
only one resulting resonant frequency. Now that the theory behind the measurement
method has been explained, the experimental apparatus, setup, and operation method is
described in the next section.
There are three components to the experimental equipment of the PASCO ESR
Apparatus: an ESR Probe Unit, an ESR Basic System, and a Complete ESR System. The
equipment comprising each of these components are seen in Figure 3 below.
Figure 3: The ESR Apparatus
The ESR Probe Unit includes: a Probe Unit with base, three RF Probes and a DPPH
sample in a vial, a Passive Resonant Circuit, and a Current Measuring Lead for the Probe
Unit. The ESR Basic System includes: an ESR Probe Unit, a pair of Helmholtz coils with
bases, and an ESR Adapter. The Complete ESR System includes: an ESR Probe Unit, a
pair of Helmholtz coils with bases, and a Control Unit. In addition to these components a
dual trace oscilloscope, a DC Ammeter, connecting wires with banana plug connectors, a
Frequency Meter, and a Low Voltage AC/DC Power Supply is also used during
experimentation. The specific details and technical data for each component is discussed
below.
The Probe Unit is the most important component of the ESR Apparatus.
Figure 4: The ESR Probe Unit
This unit allows the frequency and amplitude to be controlled with the two knobs on the
top and side of the box. However, the frequency can only be altered within the range of
values allowed by the specific RF probe. The different RF probes provide a different
range of frequencies, as displayed in Figure 3. This range of frequencies occurs because
the inductance of the probe plays a part in determining the inductance of the oscillator
circuit. The Probe Unit is plugged directly into the Control Unit when using the Complete
ESR System and plugged into an ESR Adapter when using the Basic ESR System. Next,
the Control Unit is discussed.
The Control Unit allows the Probe Unit to run and be measured with three main
functions.
Figure 5: The ESR Control Unit
First, it provides the Probe Unit and Helmholtz coils with the needed voltage power to
run. Second, it measures the RF oscillations of the Probe Unit and provides a reading on
the RF Frequency Display screen seen in Figure 5. Third, it delivers two outputs to the
dual trace oscilloscope. One output is the RF Signal Output, which is proportional to the
RF oscillator current and observes the resonance pulses. The second output is the B-Field
Output, which is proportional to the Helmholtz coils current and measures the magnetic
field. In addition to these functions, the Control Unit also has a phase shifter knob to
make adjustments for the phase shift that occurs between the Helmholtz coils’ current
and the voltage the oscilloscope reads. Next, the Helmholtz coils will be discussed.
The Helmholtz coils’ main function is to provide a uniform magnetic field for
ESR measurement. To measure the ESR the sample material is placed in between these
coils to interact with the magnetic field.
Figure 6: The Helmholtz Coils
As seen in Figure 6, the coils are placed parallel to one another by a separation equal to
the value of their radii. These coils create a magnetic field whose value is calculated from
equation (3). This discussion of the experimental components now allows the setup of the
components to be explained.
For this experimental setup, the ESR Complete System with the 30-75 MHz probe
is used. All steps of this setup are seen in Figure 7 below.
Figure 7: Setup using the Complete ESR System
First, the Helmholtz coils are placed 6 cm apart and connected to the Control Unit. The
Ammeter is then connected in series between the coils and the Control Unit. Next, the X
output of the Control Unit is connected to channel 1 of the dual trace oscilloscope. The
oscilloscope is set to a sensitivity of 1 or 2 V/div, a sweep rate of 2 or 5 ms/div, and a
coupling of DC and the Umod knob on the Control Unit is set to zero. Then the U0 knob is
set at midscale and the Umod knob is adjusted at a clockwise direction in order to increase
the AC power of the current in the Helmholtz coils. If this is done correctly, the
oscilloscope trace will show a smooth sine wave as seen in Figure 8.
Figure 8: Current in Helmholtz Coils
This sine wave occurs because the AC magnetic field covers the constant DC field.
Next, the Y output of the Control Unit is connected to channel 2 of the dual trace
oscilloscope and the oscilloscope is set to a sensitivity of 0.5 or 1 V/div and a coupling of
DC. The Control Unit is then connected to the Probe Unit and the RF Probe is placed in
the Probe Unit with the DPPH sample. After this step, the Probe Unit is flipped on and
the Amplitude knob is set to a medium value. With the Probe Unit on, the Control Unit
should provide a reading of the oscillation frequency on the frequency meter. The Probe
Unit’s Frequency Control knob is then turned until the Control Unit indicates a frequency
output around 50 MHz. Now the Umod knob is set to the 11 o’clock position at
approximately the fourth position and U0 is increased to a medium value. These settings
ensure the Helmholtz coil’s current has a value around 1 A. After these steps the
oscilloscope trace now looks like Figure 9.
Figure 9: Traces of the Oscilloscope Display
As displayed in Figure 8, the channel 1 trace demonstrates the Helmholtz coil’s current,
which is proportional to the magnetic field. In Figure 9, the channel 2 trace demonstrates
the voltage passing through the RF oscillator. The pulses in this function correspond to
the points of resonance absorption from the magnetic field. If these pulses are not seen,
the Umod and RF frequency are adjusted until they become visible. As explained above, a
phase shift occurs between the Helmholtz coils’ current and the voltage, and because of
this the traces may not be symmetrical like they are in Figure 9. To compensate for this,
the phase shifter knob is adjusted until the traces are symmetrical. Now that the setup has
been completed, the actual operation of the experiment can begin, which is described in
the next section.
To start operation, the DC current is adjusted so that the resonance pulses appear
when the AC component of the Helmholtz coils’ current is zero. When the DC current is
adjusted the resonance pulses move farther or closer to one another. This is accomplished
by adjusting the vertical position of trace 2 until the bottom part of the pulse rests on the
zero level of trace 1. When this is accomplished the traces will look like Figure 10 below.
Figure 10: Final Oscilloscope Display
With these adjustments in place, the resonance value is now the DC value indicated by
the ammeter and the frequency value is the value displayed on the Control Unit.
From the values stated above, the RF frequency and the DC current is measured.
For these frequency and current values the magnetic field is measured with a multimeter
and the highest value observed is recorded. This step is repeated for several different
values of current, and for each value the frequency and magnetic field is measured. Next,
the measured value of the magnetic field is compared to the calculated value by using
equation (3). However, the value from (3) needs to be divided by 2, to compensate for the
coils’ current being set up in parallel, since the current measured is for the total system
not just one coil. Finally, the g-factor for an electron of the DPPH sample is calculated
from equation (2). With this last calculation the operation procedure is complete. The
results acquired from this experimental operation are discussed in the next section.
The operation is repeated for five different current values. The measurements
gathered for these five different values are displayed in Figure 11 below.
Measurement
Current (A)
Frequency
Magnetic Field
Magnetic Field (mT)
(MHz)
(mT)
(Calculated)
1
0.70
36.2
1.20
1.67
2
0.80
38.4
1.34
1.92
3
0.90
39.8
1.50
2.16
4
0.99
46.6
1.73
2.37
5
1.00
41.7
1.73
2.39
Figure 11: Measurement Values
Several methods are used in order to find the value of the g-factor for an electron of
DPPH. First equation (2) is used with the measured frequency and magnetic field. The
results for this method are seen in Figure 12.
Measurement
Frequency (MHz)
Magnetic Field (mT)
g-factor
1
38.4
1.34
0.343
2
39.8
1.50
0.325
3
46.6
1.73
0.302
4
36.2
1.20
0.306
5
41.7
1.73
0.274
Figure 12: g-factor Results with Measured Magnetic Field
Next equation (2) is used with the measured frequency and calculated magnetic field. The
results for this method are seen in Figure 13 below.
Measurement
Frequency (MHz)
Magnetic Field (mT)
g-factor
(Calculated)
1
38.4
1.67
0.247
2
39.8
1.92
0.227
3
46.6
2.16
0.209
4
36.2
2.37
0.224
5
41.7
2.39
0.198
Figure 12: g-factor Results with Calculated Magnetic Field
A more accurate way of calculating the g-factor is completed by graphing Current vs.
Frequency as seen in Figure 13.
Current vs. Frequency
50
45
Frequency (MHz)
40
35
30
y = 26.544x + 17.235
25
20
15
10
5
0
0
0.2
0.4
0.6
Current (A)
0.8
1
1.2
Figure 13: Current vs. Frequency
𝜐
The value of the slope is substituted into equation (4) for 𝐼 and the resulting g-factor
value is calculated to be 0.126. However, the currently accepted value for the g-factor is
2. This discrepancy comes from errors with the experiment. These errors most likely
occur when setting up the traces on the oscilloscope and measuring the magnetic field. If
either step is not completed with pristine accuracy the precision in the g-factor
calculation is affected. Because the three methods for calculating the g-factor for this
experiment gave values only 0.1 apart, the evidence for the discrepancy stands for
measurement error.
In conclusion, the g-factor, although difficult to measure, is a very important
constant in understanding the behaviors occurring within an atom. One of these
behaviors, the resonance resulting from a magnetic field, leads to the measurement of
Electron Spin Resonance. This ESR technique allows for many factors within this
process to be observed and measured. The observations seen through the oscilloscope
provide good understanding as different factors are controlled in order to find the value of
the g-factor. This entire procedure and measurement method confirms the usefulness and
need of the Electron Spin Resonance.